The term sequence refers to the order in which specific things or occurrences occur. A sequence is a collection of integers arranged in a specified order. Sequences are used in a variety of applications, including the analysis of economic data and some fields of physics. Sequences are also crucial in the development of mathematical concepts, which have several technical applications.
Definition of Sequence
An ordered set of items or numbers, such as a1, a2, a3, a4, a5, a6,……an…., is said to be in a sequence in mathematics if it follows a given rule and has a definite value. The sequence’s members are known as terms or elements, and they can be any natural number value. Every term in a sequence is connected to the term before it and the term after it. In general, sequences feature a hidden pattern or rule that aids in determining the value of the next term.
Types of Number Sequence
Sequences are mainly divided into three categories:
- Arithmetic Sequences
- Geometric Sequence
- Triangular series
Arithmetic Sequences are any sequence in which the difference between each successive term is constant. This means that the numbers keep increasing by an arbitrary constant value as we progress up the sequence. If we need to generate the next number, we simply add this arbitrary constant value to the previous number in the sequence to generate a new number.
3, 6, 9, 12, 15, 18, 21…….
+3 +3 +3 +3 +3 +3
The difference between the two terms is 3 in this case. As a result, it’s known as the distinction.
When we need to find the next number in this sequence, we simply add 3 to the previous number.
The letter “d” stands for difference.
We can see that a1 = 3 and a2 = 6 in the example above.
The distinction between the two consecutive terms is
a2 – a1 = 3
a3 – a2 = 3
If an arithmetic sequence’s initial term is a1 and the common difference is d, the sequence’s nth term is:
We can successfully determine any number of any arithmetic sequence using the formula above.
Geometric Sequence is a sequence in which each successive term has a constant ratio between them. A constant ratio means that there is an arbitrary constant between every two numbers in a geometric sequence, which is multiplied by the sequence’s last number to get the next number.
1, 4, 16, 64…
a2 = 4 = a1(4)
a3 = 16 = a2 (4)
To get the next term, we multiply it by four every time. The ratio is 4 in this case.
“r” stands for the ratio.
a = an-1 . r or
a = a1 . rn-1
an = an-1 × r
We can calculate the number of any geometric sequence using the formula above.
The Arithmetic Series Formula
an = a + (n – 1) d is the formula for the nth term, where an is the first term, d is the difference, and n is the total number of terms.
Now, in an arithmetic series, calculate the sum to n terms. The calculation formula is listed below.
Sum of Arithmetic Series
Sn=n/2 . 2a+ (n−1)d
Sum to the nth term can be found using the formula above.
A triangular number, also known as a triangle number, is a number that counts objects in an equilateral triangle. Square numbers and cube numbers are examples of figurate numbers, as are triangular numbers. The nth triangular number is equal to the sum of the n natural numbers from 1 to n and is the number of dots in a triangular arrangement with n dots on each side. Starting with the 0th triangular number, the sequence of triangular numbers is
0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666…
Consider the following triangular series as an example:
1, 3, 6, 10, 15, 21………….
This pattern is created by filling a triangle with dots. A sequence can be obtained by adding dots to another row and counting all of the dots.
Sequences are useful in a range of mathematical disciplines… Series, which are important in differential equations and analysis, are built on the foundation of sequences. Sequences are fascinating in and of themselves, and can be examined as patterns or riddles, similar to how prime numbers are researched.